148 THEORY OF COLLINEATIONS. 
the transformations of the second kind. In particular we 
note that the determinant of T is the negative of the deter- 
minant of 7; thus T may be written 
Whence A = — A, where . is the determinant of 7. 
182. Evistence of Subgroups of G,. Our first question is 
to ask if the general projective group G, contains continuous 
subgroups. 
Let the matrix of a collineation 7 be 
a) b, Cc) || 
M=\ea b eo | : 
a; bs ¢3| 
| 
let ‘the determinant of the matrix be not zero; and let the 
characteristic equation of the matrix have three distinct roots, 
so that the collineation is of the most general type. If all the 
elements of the matrix vary independently we have the eight- 
parameter group G, of all collineations in the plane. In order 
to select out from this group a system of collineations with a 
smaller number of parameters we must reduce the number of 
independent parameters in the group G,, 7. e., we must impose 
upon the elements of Mone or more relations. This necessary 
condition is also sufficient ; for if we impose 7 relations upon 
the elements of M, we reduce by 7 the number of independ- 
ent parameters. If this system of collineations is to have the 
first group property, these relations on the elements of M 
must be of such a form that if they are imposed upon the ele- 
ments of the matrices of J and T,, they will also be satisfied 
by the elements of the matrix of T,, where T,=TT,. It 
thus appears that a necessary condition for the existence of a. 
subgroup of G, is the existence of a set of one or more rela- 
tions among the elements of M having the property that they 
are satisfied simultaneously by the elements of the matrices 
Ol Ean daelins 
